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AIP conference proceedings

Information processing in quantum spin systems

Andrew J. Landahl

Published Jan 1, 2004• 2 citations
Publisher

Abstract

We present a class of spin networks that act as perfect quantum wires—quantum states transfer across these networks with unit fidelity in a time proportional to the network’s size. No couplings need to be switched on and off during this process. We prove that N‐spin hypercube networks with equal‐strength nearest‐neighbor Heisenberg and XY couplings have a maximal perfect communication distance of 2log3N. We show how to engineer coupling strengths in linear Heisenberg and XY spin chains to efficiently achieve perfect state transfer over arbitrarily long distances. Finally, we show how to augment these linear spin chain networks with more complex, but static, interactions to achieve universal quantum computation.

DOI
10.1063/1.1834419

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We present a class of spin networks that act as perfect quantum wires—quantum states transfer across these networks with unit fidelity in a time proportional to the network’s size. No couplings need to be switched on and off during this process. We prove that N‐spin hypercube networks with equal‐strength nearest‐neighbor Heisenberg and XY couplings have a maximal perfect communication distance of 2log3N. We show how to engineer coupling strengths in linear Heisenberg and XY spin chains to efficiently achieve perfect state transfer over arbitrarily long distances. Finally, we show how to augment these linear spin chain networks with more complex, but static, interactions to achieve universal quantum computation.
We propose a class of qubit networks that admit the perfect state transfer of any quantum state in a fixed period of time. Unlike many other schemes for quantum computation and communication, these networks do not require qubit couplings to be switched on and off. When restricted to N-qubit spin networks of identical qubit couplings, we show that 2log3N is the maximal perfect communication distance for hypercube geometries. Moreover, if one allows fixed but different couplings between the qubits, then perfect state transfer can be achieved over arbitrarily long distances in a linear chain.
We propose a class of qubit networks that admit perfect state transfer of any two-dimensional quantum state in a fixed period of time. We further show that such networks can distribute arbitrary entangled states between two distant parties, and can, by using such systems in parallel, transmit the higher-dimensional systems states across the network. Unlike many other schemes for quantum computation and communication, these networks do not require qubit couplings to be switched on and off. When restricted to NNN-qubit spin networks of identical qubit couplings, we show that 2log3N2\phantom{\rule{0.2em}{0ex}}{\mathrm{log}}_{3}N2log3​N is the maximal perfect communication distance for hypercube geometries. Moreover, if one allows fixed but different couplings between the qubits then perfect state transfer can be achieved over arbitrarily long distances in a linear chain. This paper expands and extends the work done by Christandl et al., Phys. Rev. Lett. 92, 187902 (2004).
Spin chains have been proposed as wires to transport information between distributed registers in a quantum information processor. Unfortunately, the challenges in manufacturing linear chains with engineered couplings has hindered experimental implementations. Here we present strategies to achieve perfect quantum information transport in arbitrary spin networks. Our proposal is based on the weak coupling limit for pure state transport, where information is transferred between two end spins that are only weakly coupled to the rest of the network. This regime allows ignoring the complex, internal dynamics of the bulk network and relying on virtual transitions or on the coupling to a single bulk eigenmode. We further introduce control methods capable of tuning the transport process and achieve perfect fidelity with limited resources, involving only manipulation of the end qubits. These strategies could be thus applied not only to engineered systems with relaxed fabrication precision, but also to naturally occurring networks; specifically, we discuss the practical implementation of quantum state transfer between two separated nitrogen vacancy (NV) centers through a network of nitrogen substitutional impurities.
We present a communication protocol for chains of permanently coupled qubits which achieves perfect quantum state transfer and which is efficient with respect to the number of chains employed in the scheme. The system consists of M uncoupled identical quantum chains. Local control (gates, measurements) is only allowed at the sending/receiving end of the chains. Under a quite general hypothesis on the interaction Hamiltonian of the qubits, a theorem can be proved which shows that the receiver is able to asymptotically recover the messages by repetitive monitoring of his qubits. We show how two parallel Heisenberg spin chains can be used as quantum wires. Perfect state transfer with a probability of failure lower than P in a Heisenberg chain of N spin-1/2 particles can be achieved in a time scale of the order of [Formula: see text].
Quantum information transfer is an important part of quantum information processing. Several proposals for quantum information transfer along linear arrays of nearest-neighbor coupled qubits or spins were made recently. Perfect transfer was shown to exist in two models with specifically designed strongly inhomogeneous couplings. We show that perfect transfer occurs in an entire class of chains, including systems whose nearest-neighbor couplings vary only weakly along the chain. The key to these observations is the Jordan-Wigner mapping of spins to noninteracting lattice fermions that display perfectly periodic dynamics if the single-particle energy spectrum is appropriate. After a half-period of that dynamics, any state is transformed into its mirror image with respect to the center of the chain. The absence of fermion interactions preserves these features at an arbitrary temperature and allows for the transfer of nontrivially entangled states of several spins or qubits.